Classical models of aerial swarms often describe global coordinated motion as the combination of local interactions that happen at the individual level. Mathematically, these interactions are represented with Potential Fields. Despite their explanatory success, these models fail to guarantee rapid and safe collective motion when applied to aerial robotic swarms flying in cluttered environments of the real world, such as forests and urban areas. Moreover, these models necessitate a tight coupling with the deployment scenarios to induce consistent swarm behaviors. Here, we propose a predictive model that combines the local principles of potential field models with the knowledge of the agents’ dynamics. We show that our approach improves the speed, order, and safety of the swarm, it is independent of the environment layout, and scalable in the swarm speed and inter-agent distance. Our model is validated with a swarm of five quadrotors that can successfully navigate in a real-world indoor environment populated with obstacles.